a piece of wire measuring 20 feet is attached to a telephone pole as a guy wire. the distance along the ground from the bottom of the pole to the end of the wire is 4 feet greater than the height where the wire is attached to the pole. how far up the pole does the guy wire reach?
The guy wire, the telephone phone and the ground form a right-angled triangle.
Your measurements are x, x+4, and 20.
To solve for x, you could use the Pythagorean Theorem, a^2+b^2=c^2.
Let a=x
Let b=x+4
Let c=20
First, plug in the values for a, b and c, and then simplify.
From here, solve for x by either factoring, or using the quadratic formula (I'm going to use the quadratic formula).
Now, plug in the values of x into the original formula
The measurement of the telephone pole is x.
My values of x are 12 and -16.
If we substitute the values of x in,
only x=12 makes sense,
because a telephone pole with height -16 is impossible
hope it helps.
Let's call the height where the wire is attached to the pole "x". According to the given information, the distance along the ground from the bottom of the pole to the end of the wire is 4 feet greater than x.
So, the distance along the ground is x + 4.
We can form a right triangle, where the height of the pole is x and the distance along the ground is x + 4. The length of the guy wire forms the hypotenuse of this triangle.
Using the Pythagorean theorem, we can find the length of the guy wire:
Hypotenuse^2 = Height^2 + Base^2
(20)^2 = x^2 + (x + 4)^2
400 = x^2 + (x^2 + 8x + 16)
400 = 2x^2 + 8x + 16
Rearranging the equation by subtracting 400 from both sides:
0 = 2x^2 + 8x + 16 - 400
0 = 2x^2 + 8x - 384
Now let's solve the quadratic equation. We can simplify it by dividing the entire equation by 2:
0 = x^2 + 4x - 192
To factor this quadratic equation:
0 = (x - 12)(x + 16)
Setting each factor to zero:
x - 12 = 0 or x + 16 = 0
From these equations, we find two possible values for x: x = 12 or x = -16.
Since the problem refers to a physical height, we can ignore the negative value. Thus, x = 12.
Therefore, the guy wire reaches a height of 12 feet up the pole.
To solve this problem, we can set up a right triangle using the given information. Let's assume that the height where the wire is attached to the pole is 'x' feet.
According to the problem, the distance along the ground from the bottom of the pole to the end of the wire is 4 feet greater than the height. So, the distance along the ground would be 'x + 4' feet.
Using the Pythagorean Theorem, we can relate the height, the distance along the ground, and the length of the guy wire.
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In our case, the guy wire is the hypotenuse of the triangle. The length of the guy wire is 20 feet.
So, we have:
(x + 4)^2 + x^2 = 20^2
Expanding and simplifying the equation further:
x^2 + 8x + 16 + x^2 = 400
2x^2 + 8x - 384 = 0
To solve this quadratic equation, we can factor out a 2:
2(x^2 + 4x - 192) = 0
Now, we can factor the quadratic:
2(x + 16)(x - 12) = 0
Setting each factor equal to zero:
x + 16 = 0 or x - 12 = 0
Solving for 'x':
x = -16 or x = 12
Since distance cannot be negative in this context, we can disregard the solution x = -16.
Therefore, the height where the guy wire is attached to the pole is 12 feet.